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Single Idea 9995

[filed under theme 5. Theory of Logic / K. Features of Logics / 6. Compactness ]

Full Idea

If a wff is tautologically implied by a set of wff's, it is implied by a finite subset of them; and if every finite subset is satisfiable, then so is the whole set of wff's.

Clarification

a 'wff' is a well-formed formula

Gist of Idea

Proof in finite subsets is sufficient for proof in an infinite set

Source

Herbert B. Enderton (A Mathematical Introduction to Logic (2nd) [2001], 2.5)

Book Ref

Enderton,Herbert B.: 'A Mathematical Introduction to Logic' [Academic Press 2001], p.142

A Reaction

[Enderton's account is more symbolic] He adds that this also applies to models. It is a 'theorem' because it can be proved. It is a major theorem in logic, because it brings the infinite under control, and who doesn't want that?

The 17 ideas with the same theme [satisfaction by satisfying the finite subsets]:

9995 Proof in finite subsets is sufficient for proof in an infinite set [Enderton]
10771 Compactness is important for major theories which have infinitely many axioms [Tharp]
10772 Compactness blocks infinite expansion, and admits non-standard models [Tharp]
13544 Inconsistency or entailment just from functors and quantifiers is finitely based, if compact [Bostock]
13618 Compactness means an infinity of sequents on the left will add nothing new [Bostock]
13841 Why should compactness be definitive of logic? [Boolos, by Hacking]
10287 If a first-order theory entails a sentence, there is a finite subset of the theory which entails it [Hodges,W]
13496 First-order logic is 'compact': consequences of a set are consequences of a finite subset [Hart,WD]
17789 No logic which can axiomatise arithmetic can be compact or complete [Mayberry]
13630 Non-compactness is a strength of second-order logic, enabling characterisation of infinite structures [Shapiro]
13646 Compactness is derived from soundness and completeness [Shapiro]
13699 Compactness surprisingly says that no contradictions can emerge when the set goes infinite [Sider]
10975 Compactness does not deny that an inference can have infinitely many premisses [Read]
10977 Compactness blocks the proof of 'for every n, A(n)' (as the proof would be infinite) [Read]
10976 Compactness makes consequence manageable, but restricts expressive power [Read]
10974 Compactness is when any consequence of infinite propositions is the consequence of a finite subset [Read]
17867 If a concept is not compact, it will not be presentable to finite minds [Almog]